Theorem natoppfb 49236

Description: A natural transformation is natural between opposite functors, and vice versa. (Contributed by Zhi Wang, 18-Nov-2025.)

Hypotheses

Ref	Expression
natoppf.o	⊢ 𝑂 = (oppCat‘𝐶)
natoppf.p	⊢ 𝑃 = (oppCat‘𝐷)
natoppf.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
natoppf.m	⊢ 𝑀 = (𝑂 Nat 𝑃)
natoppfb.k	⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
natoppfb.l	⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
natoppfb.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
natoppfb.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
natoppfb	⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))

Proof of Theorem natoppfb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		natoppf.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
2		natoppf.p	. . . 4 ⊢ 𝑃 = (oppCat‘𝐷)
3		natoppf.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4		natoppf.m	. . . 4 ⊢ 𝑀 = (𝑂 Nat 𝑃)
5		natoppfb.k	. . . . 5 ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝐾 = ( oppFunc ‘𝐹))
7		natoppfb.l	. . . . 5 ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
8	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝐿 = ( oppFunc ‘𝐺))
9		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝑥 ∈ (𝐹𝑁𝐺))
10	1, 2, 3, 4, 6, 8, 9	natoppf2 49235	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝑥 ∈ (𝐿𝑀𝐾))
11		eqid 2729	. . . . 5 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂)
12		eqid 2729	. . . . 5 ⊢ (oppCat‘𝑃) = (oppCat‘𝑃)
13		eqid 2729	. . . . 5 ⊢ ((oppCat‘𝑂) Nat (oppCat‘𝑃)) = ((oppCat‘𝑂) Nat (oppCat‘𝑃))
14	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐿 = ( oppFunc ‘𝐺))
15	14	fveq2d 6830	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐿) = ( oppFunc ‘( oppFunc ‘𝐺)))
16	4	natrcl 17879	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐿𝑀𝐾) → (𝐿 ∈ (𝑂 Func 𝑃) ∧ 𝐾 ∈ (𝑂 Func 𝑃)))
17	16	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐿 ∈ (𝑂 Func 𝑃) ∧ 𝐾 ∈ (𝑂 Func 𝑃)))
18	17	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐿 ∈ (𝑂 Func 𝑃))
19	14, 18	eqeltrrd 2829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐺) ∈ (𝑂 Func 𝑃))
20		relfunc 17788	. . . . . . 7 ⊢ Rel (𝑂 Func 𝑃)
21		eqid 2729	. . . . . . 7 ⊢ ( oppFunc ‘𝐺) = ( oppFunc ‘𝐺)
22	19, 20, 21	2oppf 49137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘( oppFunc ‘𝐺)) = 𝐺)
23	15, 22	eqtr2d 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐺 = ( oppFunc ‘𝐿))
24	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐾 = ( oppFunc ‘𝐹))
25	24	fveq2d 6830	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐾) = ( oppFunc ‘( oppFunc ‘𝐹)))
26	17	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐾 ∈ (𝑂 Func 𝑃))
27	24, 26	eqeltrrd 2829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))
28		eqid 2729	. . . . . . 7 ⊢ ( oppFunc ‘𝐹) = ( oppFunc ‘𝐹)
29	27, 20, 28	2oppf 49137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘( oppFunc ‘𝐹)) = 𝐹)
30	25, 29	eqtr2d 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐹 = ( oppFunc ‘𝐾))
31		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐿𝑀𝐾))
32	11, 12, 4, 13, 23, 30, 31	natoppf2 49235	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐹((oppCat‘𝑂) Nat (oppCat‘𝑃))𝐺))
33	1	2oppchomf 17649	. . . . . . . 8 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))
34	33	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)))
35	1	2oppccomf 17650	. . . . . . . 8 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂))
36	35	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (compf‘𝐶) = (compf‘(oppCat‘𝑂)))
37	2	2oppchomf 17649	. . . . . . . 8 ⊢ (Homf ‘𝐷) = (Homf ‘(oppCat‘𝑃))
38	37	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (Homf ‘𝐷) = (Homf ‘(oppCat‘𝑃)))
39	2	2oppccomf 17650	. . . . . . . 8 ⊢ (compf‘𝐷) = (compf‘(oppCat‘𝑃))
40	39	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (compf‘𝐷) = (compf‘(oppCat‘𝑃)))
41		natoppfb.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐶 ∈ 𝑉)
43		natoppfb.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐷 ∈ 𝑊)
45	1, 2, 42, 44, 27	funcoppc5 49150	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐹 ∈ (𝐶 Func 𝐷))
46	45	func1st2nd 49081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
47	46	funcrcl2 49084	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐶 ∈ Cat)
48	1	oppccat 17647	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
49	11	oppccat 17647	. . . . . . . 8 ⊢ (𝑂 ∈ Cat → (oppCat‘𝑂) ∈ Cat)
50	47, 48, 49	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (oppCat‘𝑂) ∈ Cat)
51	46	funcrcl3 49085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐷 ∈ Cat)
52	2	oppccat 17647	. . . . . . . 8 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
53	12	oppccat 17647	. . . . . . . 8 ⊢ (𝑃 ∈ Cat → (oppCat‘𝑃) ∈ Cat)
54	51, 52, 53	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (oppCat‘𝑃) ∈ Cat)
55	34, 36, 38, 40, 47, 50, 51, 54	natpropd 17905	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐶 Nat 𝐷) = ((oppCat‘𝑂) Nat (oppCat‘𝑃)))
56	3, 55	eqtrid 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑁 = ((oppCat‘𝑂) Nat (oppCat‘𝑃)))
57	56	oveqd 7370	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐹𝑁𝐺) = (𝐹((oppCat‘𝑂) Nat (oppCat‘𝑃))𝐺))
58	32, 57	eleqtrrd 2831	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐹𝑁𝐺))
59	10, 58	impbida 800	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐹𝑁𝐺) ↔ 𝑥 ∈ (𝐿𝑀𝐾)))
60	59	eqrdv 2727	1 ⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ‘cfv 6486  (class class class)co 7353  1^st c1st 7929  2^nd c2nd 7930  Catccat 17589  Hom_f chomf 17591  comp_fccomf 17592  oppCatcoppc 17636   Func cfunc 17780   Nat cnat 17870   oppFunc coppf 49127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-map 8762  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12611  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17140  df-hom 17204  df-cco 17205  df-cat 17593  df-cid 17594  df-homf 17595  df-comf 17596  df-oppc 17637  df-func 17784  df-nat 17872  df-oppf 49128

This theorem is referenced by:  fucoppclem  49412  lmddu  49672